# Characteristics of a Hyperbola

Hyperbolas are conic sections formed when a plane intersects a pair of cones. Hyperbolas have the characteristic that the difference of the distances from any point on the curve to the two foci is equal to a constant. Hyperbolas are made up of two branches that have a parabolic shape. All hyperbolas have two lines of symmetry, which intersect at the center.

Here, we will look at a more detailed definition of hyperbolas and we will learn about some of their most important characteristics.

##### PRECALCULUS

Relevant for

Learning about the fundamental characteristics of hyperbolas.

See characteristics

##### PRECALCULUS

Relevant for

Learning about the fundamental characteristics of hyperbolas.

See characteristics

## Definition of a hyperbola

A hyperbola is defined as the set of points so that the difference of the distances to the two foci is a constant. Hyperbolas are made up of two branches, which are a reflection of each other. Each branch of the hyperbola is similar to a parabola and has a focus and a vertex.

Hyperbolas are also defined as conic sections that are obtained at the intersection of a plane with a pair of cones. The plane cuts both bases of the cones at a certain angle.

## Main characteristics of a hyperbola

The main characteristics of a hyperbola are:

• Hyperbolas have two focal points, called foci.
• The eccentricity of the hyperbolas is greater than 1.
• The difference of each distance from a point on the hyperbola to the two foci is constant.
• Hyperbolas have two axes of symmetry, one axis passes through the foci and the other axis is perpendicular to the first.
• The intersection of the lines of symmetry is the center of the hyperbola.
• Hyperbolas have two asymptotes, towards which they approach, but never touch.
• The asymptotes also intersect at the center of the hyperbola.

## Equation of a hyperbola

The form of the hyperbola equation depends on whether the hyperbola is centered at the origin or centered outside the origin.

When we have a hyperbola centered on the origin, its general equation is:

where a represents the length of the segment that extends between the two vertices of the hyperbola and b is found with the equation $latex {{b}^2}={{a}^2}({{e}^2}-1)$, where, e is the eccentricity.

If the center of the hyperbola is located outside the origin, the equation of the hyperbola is:

where, $latex (h, k)$ are the coordinates of the center of the hyperbola.  